_ n (1^2-1 )(n-1)+ (2^2-2 )(n-2)+ + ((n-1)^2-(n-1) ) 1 (1^3+2^3+ +n^3 )- (1^2+2^2+ +n^2 ) is equal to :

Step-by-step solution

aligned & _ n _ r=1 ^ n-1 (r^2-r )(n-r) _ r=1 ^n r^3- _ r=1 ^n r^2 & _ n _ r =1 ^ n -1 (- r ^3+ r ^2( n +1)- nr ) ( n ( n +1) 2 )^2- n ( n +1)(2 n +1) 6 & _ n ( ((n-1) n) 2 )^2+ (n+1)(n-1) n(2 n-1) 6 - n^2(n-1) 2 n(n+1) 2 ( n(n+1) 2 - 2 n+1 3 ) & _ n n(n-1) 2 ( -n(n-1) 2 + (n+1)(2 n-1) 3 -n ) n(n+1) 2 3 n^2+3 n-4 n-2 6 & _ n (n-1) (-3 n^2+3 n+2 (2 n^2+n-1 )-6 ) (n+1) (3 n^2-n-2 ) & _ n (n-1) (n^2+5 n-8 ) (n+1) (3 n^2-n-2 ) = 1 3 & aligned